The Arm/Cartridge Matching Myth

The point that I was trying to make is that when an input frequency dwells on the resonance point of a mechanical system, the output amplitude tries to go to infinity. Of course, there are limitations but usually the limitation is failure of the system- something breaks. Picture Ella Fitzgerald focusing her voice on a crystal goblet. She finds the resonance frequency of the goblet and then sings holding that frequency steady. The glass responds to that input. The goblet starts to ring and the amplitude builds until the glass shatters. All with the relatively small energy input of her voice.

Harold, there is no single equation that I am aware of for determining the effective mass of the tonearm in grams. It is an involved calculation based on the mass of the tonearm and the moment of inertia of the tonearm assembly which, as you said must be broken down for both sides of the fulcrum. Also, the effective mass is not likely to be the same value in the lateral direction (unless the tonearm and counterweight are straight cylinders), just like the cartridge compliance will not be the same value in the lateral direction. That just means the dynamic response laterally will be different from the vertical dynamic response. And there again, the static forces should be the same in both the horizontal and vertical planes. I can't think of a reason why they wouldn't be.

I hope that I can clarify a few things for everyone. Resonance is a phenomena where a small amount of energy in results in near infinite energy out. It can occur in either mechanical or electrical systems. Static forces and dynamic frequency response are independent of each other. ie. Tracking force does not affect the natural frequency of the cartridge/tonearm system- unless you have so much tracking force that the cartridge suspension is bottomed out. You'll know that is the case if you see a thin ribbon of vinyl curling up behind the needle while playing a record.
Effective Mass is a shortcut for calculating the tonearm/cartridge system natural frequency. Effective mass is really the inertia of the tonearm expressed in grams. That's because the tonearm is resting on a fulcrum (pivot). The tonearm and counterweight weigh much more than just a few grams. Additionally, the effective mass can vary depending on the position of the counterweight. So a lower mass cartridge will lower the effective mass of the tonearm since the counter weight will be closer to the pivot. I guess the tonearm manufactures provide us with a nominal value. Don't forget to add the mass of the cartridge, not just the spring rate to the natural frequency calculation.
The only magic about 10Hz +/-2 Hz is that this is the "quiet" area. Below 8 Hz is the range of record warps and footfalls. Above 16 Hz is getting into the range of the music. The tonearm/cartridge system is still responding at 10Hz. Any energy input will make it respond at that frequency. The key is not to have input at 10Hz. This is so the inherent damping of the stylus suspension and any tonearm pivot bearing friction can be effective at keeping the tonearm/cartridge calm. That small amount of damping keeps things under control. If the tonearm system has a response at say 3-5Hz then the resonance- infinite energy out will overwhelm the damping properties and the tonearm will be greatly excited when rising over a record warp. You could certainly not have a problem with a tonearm system response of 5Hz as long as the records are perfectly flat, hole dead center and the turntable well isolated from footfalls.
I had a cartridge/tonearm system at 16 Hz once. It sounded ok but when I looked closely, the stylus was constanltly moving up and down. I rectified the problem by adding a 4 gram mass to the headshell. That dropped the natural frequency to about 10Hz. Everything was steady then.
The danger is that being outside the quiet zone (10Hz +/-) can result in excessive wear or even damage to your cartridge and records- even if it sounds fine.

Hi Halcro, I gave one example from a personal experience. The tonearm/stylus was cycling at 16 Hz while playing a record. It was interesting to see. Long term it would have worn out the stylus for sure.
The other thing that I have seen happen before is the stylus jump the groove with only the slightest provocation.
That makes me think of something: Is it only me or do others try not to cue the needle in the middle of the record? I have this feeling that dropping the needle, even ever so slowly in the middle of the record will leave some minor damage in the groove. So I tend to cue at the beginning and let it play through to the track that I want to hear. Is that a pointless exercise?